Bring a medium saucepan of water to boiling. Add cauliflower, squash, mushrooms, and 1 sprig thyme; cover and return to boiling. Remove pan from heat. Using a slotted spoon, transfer vegetables to a bowl; remove and reserve thyme sprig. Reserve cooking liquid in pan; cover to keep hot.

Preheat oven to 400°F. Place onions in an 8-inch square glass baking dish; add the remaining 3 sprigs thyme and ½ cup of the hot cooking liquid. Bake 1 hour or until onions are a rich dark brown, stirring every 15 minutes and adding ¼ to ½ cup cooking liquid whenever the mixture begins to look dry. If necessary, cook 5 to 10 minutes more to cook off any excess liquid. Transfer onion mixture to a bowl; discard thyme sprigs. Gently stir in blanched vegetables. Let cool 10 to 15 minutes.

Meanwhile, in a medium bowl whisk together flour and salt. Rub tahini into flour with a fork until it resembles coarse sand. Stir in ⅓ to ½ cup of the reserved cooking liquid until a smooth, firm dough forms. Let dough cool 10 minutes.

Line a baking sheet with parchment paper. Roll out dough to a 10-inch disk. Transfer disk to the prepared baking sheet.

Spread vegetable mixture in the center of disk, leaving a 2-inch border. Fold the dough up and around the vegetables to create a raised edge, pleating as you go. Bake 30 to 35 minutes or until crust is browned and crisp. Garnish with the reserved thyme sprig.